I am a patient... But I am can pretty much guess that the nose bleeds were side effects from the blood thinners and he was taking those because of his heart surgery.
Brain tumors can have few or many symptoms. My cousin who recently passed did not know about his until one morning he woke up and was not speaking right. Never had a headache.
The brain is well separate from the nasal area and it would take a heck of a tumor to break through the dura (the sheath around the brain) and go through bone to then make nose bleeds. He would be in sorry shape.
Sorry for your loss.